LAUSR

We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schrodinger operators each of which has a simple well potential. Focusing on the cases where the two underlying classical periodic trajectories cross to each other, we give Bohr–Sommerfeld type quantization rules for the eigenvalues of the system in both tangential and transversal crossing cases. Our main results consist in the eigenvalue splitting which occurs when the two action integrals along the closed trajectories coincide. The splitting is of polynomial order $$h^{{\frac{4}{3}}}$$ in the tangential case and of order $$h^{{\frac{3}{2}}}$$ in the transversal case, and the coefficients of the leading terms reflect the geometry of the crossing.